Don't know about the tonkatsu sauce, but 中農 sauce is just fine outside the fridge. In fact, it seems to get better the longer it is left out. I always like to pour a little bit of the old bottle into the new bottle just to help get it going. Sort of like sourdough starter mash, I suppose.

There was a liquor store I used to live at ... er, near... and they had white lightning in MASON JARS with butcher-paper labels on it saying "Corn Squeezins -- product of Kentucky, guaranteed less than one month old." -- great stuf <cough cough... Smooth... cough>